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Properties

Label 2-280-40.3-c1-0-26
Degree $2$
Conductor $280$
Sign $0.406 + 0.913i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.341i)2-s + (−1.37 − 1.37i)3-s + (1.76 − 0.937i)4-s + (1.91 + 1.15i)5-s + (−2.36 − 1.42i)6-s + (−0.707 − 0.707i)7-s + (2.10 − 1.89i)8-s + 0.798i·9-s + (3.01 + 0.938i)10-s + 1.20·11-s + (−3.72 − 1.14i)12-s + (0.687 − 0.687i)13-s + (−1.21 − 0.728i)14-s + (−1.03 − 4.23i)15-s + (2.24 − 3.31i)16-s + (−5.48 + 5.48i)17-s + ⋯
L(s)  = 1  + (0.970 − 0.241i)2-s + (−0.795 − 0.795i)3-s + (0.883 − 0.468i)4-s + (0.854 + 0.518i)5-s + (−0.964 − 0.579i)6-s + (−0.267 − 0.267i)7-s + (0.743 − 0.668i)8-s + 0.266i·9-s + (0.954 + 0.296i)10-s + 0.361·11-s + (−1.07 − 0.329i)12-s + (0.190 − 0.190i)13-s + (−0.323 − 0.194i)14-s + (−0.267 − 1.09i)15-s + (0.560 − 0.828i)16-s + (−1.33 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65264 - 1.07360i\)
\(L(\frac12)\) \(\approx\) \(1.65264 - 1.07360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.341i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.37 + 1.37i)T + 3iT^{2} \)
11 \( 1 - 1.20T + 11T^{2} \)
13 \( 1 + (-0.687 + 0.687i)T - 13iT^{2} \)
17 \( 1 + (5.48 - 5.48i)T - 17iT^{2} \)
19 \( 1 + 7.61iT - 19T^{2} \)
23 \( 1 + (1.47 - 1.47i)T - 23iT^{2} \)
29 \( 1 - 6.22T + 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (-3.15 - 3.15i)T + 37iT^{2} \)
41 \( 1 + 1.94T + 41T^{2} \)
43 \( 1 + (2.24 + 2.24i)T + 43iT^{2} \)
47 \( 1 + (-2.02 - 2.02i)T + 47iT^{2} \)
53 \( 1 + (4.92 - 4.92i)T - 53iT^{2} \)
59 \( 1 - 8.54iT - 59T^{2} \)
61 \( 1 - 1.59iT - 61T^{2} \)
67 \( 1 + (-0.740 + 0.740i)T - 67iT^{2} \)
71 \( 1 - 1.77iT - 71T^{2} \)
73 \( 1 + (5.23 + 5.23i)T + 73iT^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + (3.01 + 3.01i)T + 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + (-10.5 + 10.5i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.77575501817544175115612527297, −10.93957570882384345363316927932, −10.29403234587403059602526984285, −8.916265559011553373317759251274, −7.04200380716321375223377965003, −6.58277154869503967108058551342, −5.85766148518272265245358229065, −4.56993288314163524589904792114, −2.98375349933066283103512983612, −1.51123985039089546501397065661, 2.27585118634984628864010079714, 4.08119237761222197729398753571, 4.93234681861278456393413582393, 5.86678553159591832504495302250, 6.54041669994133575418780568647, 8.120371634675629667482115176769, 9.428201285825678269481959804451, 10.25207032891042657516108590164, 11.31781867161455232021791778127, 11.99076823778630181614751612174

Graph of the $Z$-function along the critical line